Bias-correction schemes for calibrated flow in a conceptual hydrological model

We explore post-processing methods that can reduce biases in simulated flow in a hydrological model (HYMOD). Here, three bias-correction methods are compared using a set of calibrated parameters as a baseline (Cases 1 and 5). The proposed bias-correction methods are based on a flow duration curve (Case 2), an autoregressive model based on residuals obtained from simulated flows (Case 3), and a rating curve (Case 4). A clear seasonality representing a more substantial variability in winter than summer was evident in all cases. The extended range of residuals was usually observed in winter, indicating that the HYMOD may not reproduce high flows appropriately. This study confirmed that bias-corrected flows are more effective than the baseline model in terms of correcting a systematic error in the simulated flow. Moreover, a comparison of root mean square error over different flow regimes demonstrates that Case 3 is the most effective at correcting systematic biases over the entire flow regime. Finally, monthly water balances for all cases are evaluated and compared during both calibration and validation periods. The water balance in Case 3 is also closer to the observed values. The effects of different post-processing approaches on the performance of bias-correction are examined and discussed.


INTRODUCTION
Hydrological models are essential tools (Madsen ; Wagener et al. ) for understanding water cycles that provide a basis for decision-making related to water resource management, land planning, infrastructure design, and water supply. They are also imperfect representations of the real world due to model uncertainties, measurement errors, sampling errors stemming from insufficient data, climate variability, and inadequate descriptions of initial and boundary conditions. Understanding the hydrological behavior of a catchment is a complicated task (De Vos et al. These studies suggest that post-processing can improve a conventional calibration approach in an efficient way (Shi et al. Previous studies have focused separately on the hydrological model calibration process and the statistical post-processing approach. This study took an integrated approach that combined post-processing techniques (i.e., bias-correction methods) with the calibration of a hydrological model. Specifically, three bias-correction models (Cases 2, 3, and 4) were proposed to reduce systematic biases in streamflow simulation. The proposed bias-correction approaches were evaluated along with a conceptual hydrological model. To the best of our knowledge, few studies have combined hydrological model calibration and bias-correction approaches in a single model. Model performance was also compared with simulated flow obtained from an existing calibration approach (Case 1) and a subannual calibration approach (Case 5). A brief overview of the post-processing procedures considered in this study follows: • Case 1 was based on the use of an existing calibration approach without bias correction.
• Case 2 was based on flow duration curves (FDCs) of observed and calibrated flows.
• Case 3 was based on an autoregressive model constructed from time series of the residual, which were calculated from observed and simulated flows.
• Case 4 was also based on an autoregressive model constructed from time series of the residual. However, the residual was calculated from observed and simulated river levels, which were converted using a rating curve.
• Case 5 was based on a sub-annual calibration approach without bias correction. Two different parameter sets are obtained by calibrating the dry season (spring and summer) and the wet season (autumn and winter) separately.
We compared the performance of the simulated flow from a well-calibrated model as a baseline (Cases 1 and 5) with the three bias-correction models (Cases 2, 3, and 4) to address the following questions: (1) Can a seasonal pattern and systematic trend in simulated flow from a rainfall-runoff model be identified?
(2) Can a bias-correction approach reduce systematic biases in the residual for both calibration and validation periods? What is the most effective method of correcting biases of an entire flow regime?
(3) Can simulated flow with bias correction effectively reproduce observed water balance?
The balance of this study is organized as follows. In the section 'Case Study Area and the Hydrological Model', we describe the study area, data, and hydrological model. The three different post-processing methods introduced in this study are summarized in the section 'Bias-Correction Scheme'. The results of different bias-correction methods are presented in the 'Results and Discussion'. The main conclusions and discussions of this study are provided in the section 'Discussion and Conclusion'.

CASE STUDY AREA AND THE HYDROLOGICAL MODEL Study area and data
The Thorverton catchment has an area of 606 km 2 and is a sub-catchment of the Exe catchment. The Exe catchment is in the southwest of England and has an area of 1,530 km 2 and an average annual rainfall of 1,088 mm.    Table 1, and the model structure is illustrated in Figure 3. The cumulative distribution function of the water storage capacity, C, has the following form: where C max is the maximum soil moisture storage capacity in the catchment, and b exp controls the degree of spatial More details can be found in Tolson & Shoemaker ().
Calibration was conducted to obtain an optimal set of parameters with an objective function based on the RMSE by minimizing the difference between the observed and simulated flows, and the optimized parameters were tested in the validation period by: where Q sim and Q obs are the simulated and observed runoff, respectively. i is the ith day, and N is the number of days in the calibration period. The effect of different objective functions and their combinations in the calibration of rainfallrunoff models was not explored.   • First, the FDC was built with data from the entire calibration period for both observed and simulated flows.

BIAS-CORRECTION SCHEME
• Second, the probability of the simulated flow (FDC sim (Q sim )) was computed from the simulated FDC.
• Third, the bias-corrected flow (Q corr sim ) was computed by mapping the probability of simulated flow obtained from the simulated FDC onto the observed FDC (FDC obs (Q obs )). The equation can be expressed as follows: where FDC sim is constructed from the simulated flow, FDC À1 obs is an inverse function built with the observed flow, and Q corr sim is the bias-corrected flow. The FDC played the role of a transfer function for bias correction of the simulated flow by matching the FDC of observed flow for the same period. The transfer functions were estimated for 30 years , and these transfer functions were then applied to the remaining period (1991)(1992)(1993)(1994)(1995)(1996)(1997)(1998)(1999)(2000)(2001)(2002)(2003)(2004)(2005)(2006)(2007)(2008) for validation.
The following bias-correction process is based on a firstorder autoregressive model of the residual AR Q (1), named Case 3: • First, the residual series was obtained by subtracting the simulated flow from that of the observed (Equation (4)).
• Second, the residual was assumed to follow the first-order autoregressive model, AR Q (1), as written in Equation (5).
• Third, the bias-corrected flow was obtained by adding the modeled residuals from AR Q (1) to the simulated flow (Equation (6)) as follows: where t is time index (days) and T is the number of records. ε tþ1 is the modeled residual at time t þ 1, while a and b are coefficients of the AR Q (1) model.
In the proposed bias-correction method (Case 4), AR H (1) was applied in the rating curve domain. In other words, the simulated flow was mapped to the rating curve to obtain the water stage instead of flow. Moreover, the residual was assumed to follow the first-order autoregressive model as adopted by Case 3. The key aspects of the biascorrection methods in Case 4 can be summarized as follows: • First, the simulated flow was converted to the water stage through a rating curve.
• Second, the residual sequences were obtained by subtracting the simulated water stage from that of the observed value (Equation (7)).
• Fourth, the bias-corrected water stage was synthesized by adding the modeled residuals from AR H (1) to the simulated water stage (Equation (9)).
• Fifth, the bias-corrected flow was obtained by mapping the water stage onto the rating curve (Equation (10)).
Kim & Han () explored sub-annual calibration schemes. They found that the model calibrated on the subannual period schemes generally performs better than the model calibrated on the entire datasets without its separation. Therefore, the sub-annual calibration method is additionally considered as a reference (Case 5) for comparison with those of the bias-correction schemes (Cases 2, 3, and 4). The purpose of this experiment was to explore whether an improved calibration framework can be more effective than a post-processing approach to the simulated flow. More specifically, the model (

RESULTS AND DISCUSSION
Comparison of results during the calibration period   'Max-Min' represents a range between the maximum value and the minimum value. 'interquartile' represents lower and upper quartiles (25th and 75th percentiles). 'RR' represents a reduction ratio of residuals for Cases 2, 3, 4 and 5 against Case 1.
negative bias was evident in Cases 2, 3, and 4. However, negative biases ranging from À40 to À60 were slightly increased by the use of the bias-correction approach, which is depicted in Figure 6. The slight increase in negative residual in a certain range can be attributed to the shift from a positive bias after bias correction, but the shift toward the negative reported here was not significant.     Table 3 Table 4 presents the monthly water balances of the simulated flows for all four cases during the calibration period.
The water balance of Case 3 was closer to that of the observed value, in line with the results presented in Figure 7 and Table 3. The water balance for Case 2 on an annual basis was equal to the observed value because bias correction was based on an FDC. However, monthly differences in water balance were substantially increased. Apart from Case 2, Case 3 was the most similar to the observed values.

Results during the validation period
As illustrated in Figure 5 during the calibration period,  basis. This may be because the temporal pattern of the bias-

DISCUSSION AND CONCLUSIONS
The FDC model effectively reduced biases during the calibration period but was ineffective with respect to both RMSE and water balance during the validation period. As illustrated in Figure   Hydrological models are imperfect representations of real hydrological circulation due to incomplete model structure, insufficient data, measurement errors, and inadequate initial and boundary conditions. The main objective of this study was to assess whether post-processing methods can reduce the biases in hydrological model outputs. Results of three bias-correction methods were compared with the conventionally calibrated flow and sub-annually calibrated flow.
The main conclusions can be summarized as follows: 1 3. This study confirmed that the bias-corrected flows with three cases were more effective than the baseline model 5. An AR model (Cases 3 and 4) showed relatively good efficiency than the conventional calibration method (Case 1) and the FDC model (Case 2) at correcting systematic biases both in calibration and validation periods. Here, the AR model is only applicable when the lagged residual is obtained from observations at the previous time step.
For example, the AR model cannot be applied to correct the simulated flow in the future period (e.g., 2050-2100).
To alleviate this issue, the non-stationary nature of the FDC needs to be explored for better representation of time-varying behavior in the residual, including the identification of the optimal temporal scales for the construction of an FDC curve within an optimization framework. Moreover, future research should include a rigorous comparison of the bias-correction approach under different conditions (e.g., different hydrological models, catchment size, climatic conditions, land use, and terrain) to provide a general guideline.
6. A sub-annual calibration scheme (Case 5) was introduced to explore whether an improved calibration framework can be more effective than the post-processing approach to the simulated flow. The results showed that this model performance did not show much improvement compared with three post-processing approaches.